In Efron's 1970 batting-average example, James-Stein shrinkage roughly halved the total squared error of predicting rest-of-season performance

Efron & Hastie's Computer Age Statistical Inference illustrates the abstract James-Stein domination result with a concrete, computable table rather than an asymptotic argument. Taking 18 Major League Baseball players' 1970 season, each player's batting average over his first 90 at-bats serves as the raw maximum-likelihood estimate of his true underlying skill; the remainder of the season serves as the "truth" being predicted. Shrinking each player's early average toward the group's grand average produces the James-Stein prediction.

The reported outcome: "Sum of squared errors for predicting TRUTH: MLE .0425, JS .0218" — the shrinkage estimator incurs roughly half the total squared prediction error of the individual averages, across the 18-player cohort, despite the players' performances being mutually unrelated. The effect is not asymptotic hand-waving; it is a specific number produced from a specific 1970 dataset — with one caveat from the source itself: the chapter's footnote states the data "is based on 1970 Major League performances, but is partly artificial; see the end notes" (added at audit 2026-07-12).

The example is why the theorem became culturally legible: batting average is a familiar quantity, and "pool the rookies toward the league average and you predict better" is an intuition a non-statistician can carry. It concretizes the same "borrow strength from unrelated data" mechanism that links the result to Robbins-era empirical Bayes (claim-robbins-monro-1951-stochastic-approximation).